Simple Separable Diff EQ?

Solve: dy/dx = y/x

My issue not the diff eq itself, but that all solutions say the result is y = Cx. Shouldn't it be that |y| = C|x|, because the antiderivative of any equation of the form 1/u is ln|u|, not simply ln(u)? At what point can we simply get rid of the absolute value signs?